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Understanding the local flow rate peak of a hopper discharging discs through an obstacle using a Tetris-like model

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Placing a round obstacle above the orifice of a flat hopper discharging uniform frictional discs has been experimentally and numerically shown in the literature to create a local peak in the gravity-driven hopper flow rate. Using frictionless molecular dynamics simulations, we show that the local peak is unrelated to the interparticle friction, the particle dispersity, and the obstacle geometry. We then construct a probabilistic Tetris-like model, where particles update their positions according to prescribed rules rather than in response to forces, and show that Newtonian dynamics is also not responsible for the local peak. Finally, we propose that the local peak is caused by an interplay between the flow rate around the obstacle, greater than the maximum when the hopper contains no obstacle, and a slow response time, allowing the overflowing particles to achieve a higher local area packing fraction by converging well upon reaching the hopper orifice.

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GJG gratefully acknowledges financial support from startup funding of Shizuoka University (Japan).

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Correspondence to Guo-Jie Jason Gao.

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The authors declare that they have no conflict of interest. The research presented did not involve human participants and/or animals.

Appendix: frictionless MD simulations

Appendix: frictionless MD simulations

1.1 System geometry

In our MD simulations studying the gravity-driven discharging of monodisperse or 50–50 bidisperse frictionless circular dry particles, shown schematically in Fig. 8, the hopper and the obstacle have the same geometry as in the Tetris-like model. The obstacle could be one disc of diameter D or three horizontally-aligned discs, each of diameter D / 3 to resemble a flat obstacle. The disc diameter d of the monodisperse system is about the same as the large disc diameter \(d_l\) of the bidisperse system, with \(L/d=83\) and \(L/d_l=82.857\). The size ratio between the obstacle and a particle is \(D/d=9.296\) and \(D/d_l=9.28\). In the bidisperse system, the diameter ratio between large and small discs is \(d_l/d_s=1.4\) to prevent artificial crystallization in a two dimensional environment. There are N discs in the system, where \(N = 2048\) and 2712 for the monodisperse and bidisperse systems, respectively. These values ensure that the particles in each system only fill the hopper up to about 2 / 3 of its height while a steady hopper flow is maintained. To maintain a constant number of particles N in our hopper flow simulation, a particle dropping out of the hopper will reenter it from its top border by artificially shifting the particle’s vertical (y) position by a distance L while keeping its horizontal (x) position and velocities in both directions unchanged.

Fig. 8
figure 8

(Color online) The MD simulation setup modeling steady gravity-driven hopper flow of frictionless discs (blue circles) of diameter d. An obstacle (green circle) of diameter D sits at a height H above the orifice of a symmetric hopper (green straight lines) with a height L and a hopper angle \(\theta \). Gravity g is in the downward (− y) direction. The inset shows that the discs are subject to interparticle normal forces only

1.2 Interactions between objects within the system

In our MD simulation, initially orderly placed discs fall under gravity and the system eventually reaches a steady state to form a gravity-driven hopper flow. Each particle i obeys Newton’s translational equation of motion

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^{\mathrm{int}}} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^W} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i} ^I} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^G} = {m_i}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{a}{{}_{i}} ,$$

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}}\) is the total force acting on particle i with mass \(m_i\), and acceleration \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{a}{{}_{i}}\). \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^{\mathrm{int}}} \), \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^W} \), \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^I} \) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^G} \) are forces acting on particle i from its contact neighbors, the hopper wall, the obstacle, and gravity, respectively.

The simplest model of frictionless granular materials considers only the interparticle normal forces [16]. The interparticle force \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^{\mathrm{int}}} \) on particle i having \(N_c\) contact neighbors can be expressed as

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_{i}^{\mathrm{int}}} = \sum \limits _{j \ne i}^{{N_c}} {[\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{f}{{}_{ij}^n} ({r_{ij}}) + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{f}{{}_{ij}^{d}} ({r_{ij}})]}, $$

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{f}{{} _{ij}^n} ({r_{ij}})\) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{f}{{} _{ij}^d} ({r_{ij}})\) are the interparticle normal force and normal damping force defined below in Eqs. (5) and (6), respectively.

Specifically, we assume that each frictionless particle i is subjected to a finite-range, purely repulsive linear spring normal force from its contact neighbor j

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{f}{{} _{ij}^n} ({r_{ij}}) = \frac{\epsilon }{{d_{ij}^2}}{\delta _{ij}}\Theta ({\delta _{ij}}){{{\hat{r}}}_{ij}}, $$

where \(r_{ij}\) is the separation between disc particles i and j, \(\epsilon \) is the characteristic elastic energy scale, \(d_{ij} = (d_i+d_j)/2\) is the average diameter, \(\delta _{ij}=d_{ij}-r_{ij}\) is the interparticle overlap, \(\Theta (x)\) is the Heaviside step function, and \({{{\hat{r}}}_{ij}}\) is the unit vector connecting particle centers.

Similarly, we consider only the interparticle normal damping force proportional to the relative velocity between particles i and j

$$\begin{aligned} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{f}{{} _{ij}^d} ({r_{ij}}) = - b\Theta ({\delta _{ij}})(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{v} {{}_{ij}} \cdot {{{\hat{r}}}_{ij}}){{{\hat{r}}}_{ij}}, \end{aligned}$$

where b is the damping parameter, and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{v} {{}_{ij}} \) is the relative velocity between the two particles. The normal damping force results in deduction of the kinetic energy of the system after each pairwise collision.

The interaction force \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{} _i^W} \) between particle i and a hopper wall has an analogous form to the interparticle interaction \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{} _i^{\mathrm{int}}} \) with \(\epsilon ^W=2\epsilon \), which means when a particle hits a wall, it experiences a repulsive force as if it hit another mirrored self on the other side of the wall. The particle-obstacle interaction force \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_i^I} \) also has analogous form, and its value stays zero if the hopper contains no obstacle. Finally, \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F}{{}_i^G} = - {m_i}g{\hat{y}}\), where g is the gravitational constant, and \({\hat{y}}\) is the unit vector in the upward direction. There is no tangential interaction on particles in this model, and therefore Newton’s rotational equation of motion is automatically satisfied.

The MD simulations in this study use the diameter d and the mass m of the monodisperse particles and the interparticle elastic potential amplitude \(\epsilon \) as the reference length, mass, and energy scales, respectively. For the bidisperse system, the diameter \(d_s\) and the mass \(m_s\) of the small particles separately replace d and m. To maintain a steady hopper flow without particles piling up to the upper border of the hopper and bringing in unwanted boundary effects, we use the dimensionless damping parameter to \(b^*=db/\sqrt{m\epsilon }=0.5\), the dimensionless gravity \(g^*\) to \(10^{-4}\), and a dimensionless time step \(dt^*=dt/{d}\sqrt{{m}/\epsilon }\) to \(10^{-3}\) throughout this study.

1.3 Measuring the hopper flow rate

To measure the hopper flow rate while the obstacle is placed at a given value of H above the hopper orifice, we initiate one simulation with orderly arranged particles. We also randomized size identities for the bidisperse system. Then we wait for a time interval \(\Delta t^* = 5\times 10^4\) until the system forgets the initial arrangement and reaches a steady state to form a gravity-driven hopper flow. After that, we count the number of particles passing the orifice of the hopper within another \(\Delta t^*\). For each value of H, we use 18 different initial conditions to evaluate the average and the variance of the actual flow rate \(J_a\) in terms of number of particles leaving the hopper per unit time. We define \(J_o\) as the value of \(J_a\) while the hopper contains no obstacle.

1.4 Simulation results

Our investigation contains two parts: (A) To understand the influence of the interparticle friction on the locally enhanced hopper flow rate, we compare our frictionless results of the same hopper geometry with the frictional data, copied from reference [5], where monodisperse disc particles are passing about a round obstacle. (B) To understand the contribution of the obstacle geometry or particle dispersity, we measured the flow rates of frictionless discs in three cases: (1) monodisperse discs and a round obstacle, (2) monodisperse discs and a flat obstacle, and (3) 50–50 bidisperse discs and a round obstacle. The results are shown in Fig. 9, where the actual flow rate \(J_a\), normalized by \(J_o\), is plotted against the normalized obstacle position H/d or \(H/d_l\) for the monodisperse or bidisperse system. \(J_o\) is \(\approx 0.0319\) for the monodisperse system, and its value increases by about \(27\%\) to \(\approx 0.0406\) for the bidisperse system.

Fig. 9
figure 9

(Color online) Averaged frictionless flow rates \(J_a/J_o\) under different simulation setup: (1) monodisperse discs and a round obstacle (red); (2) monodisperse discs and a near flat obstacle (green); (3) bidisperse discs and a round obstacle (blue). The inset zooms in the dashed area. Each data point is obtained using 18 different initial conditions. A simulation snapshot of each frictionless setup is shown on the top with corresponding border color. The frictional flow rate of monodisperse discs and a round obstacle (black) is reproduced from Fig. 3b in Ref. [5] for a quantitative comparison

1.4.1 Comparing with the frictional data

Unlike their frictional counterparts, reproduced from reference [5], frictionless particles start to flow earlier and the normalized flow rate \(J_a/J_o\) already reaches about \(60\%\) or higher as the obstacle is lifted to about ten-particles high (\(H/d \approx 10\)) above the hopper orifice. On the other hand, the frictional normalized flow rate is only slightly above zero at a similar H/d. The local peak value of frictional \(J_a/J_o\) can be greater than unity, while all three frictionless peaks have \(J_a/J_o\) below unity with lower heights.

1.4.2 Comparing between frictionless cases

We find that the normalized hopper flow rate \(J_a/J_o\) exhibits a local peak in all three frictionless cases when the obstacle is lifted to about eleven to twelve particles high (\(H/d \approx 11\) to 12) above the orifice of the hopper. Among the three cases, the bidisperse one with a larger \(J_o\) exhibits its flow rate peak at \(H/d \approx 11.4\), earlier than the other two monodisperse cases. Between the two monodisperse cases with a round and a flat obstacle, the round obstacle blocks the hopper flow less than the flat one, and the system shows a peak slightly earlier at \(H/d \approx 12\).

1.4.3 An necessary condition for the local flow rate peak

Our results clearly show that none of the interparticle friction, the obstacle geometry, or the particle dispersity is directly responsible for the appearance of a local flow rate peak, though they do effectively affect its position and magnitude. To better predict when a flow rate peak occurs, we propose an indicator which is the flow rate \(J_i\), measured at the obstacle and normalized by \(J_o\) while the hopper contains no obstacle, as schematically shown in Fig. 10a. Here we measure \(J_i\) at the same vertical height where the center of the obstacle is located, that is, the height H above the orifice of the hopper. Practically, we measure \(J_i\) by cutting off the part of the hopper below the center of the obstacle so that the removed piece of hopper has no effect on \(J_i\).

Fig. 10
figure 10

(Color online) a1 Schematic defining a fluidized flow regime where the flow rate \(J_i\) at the obstacle is smaller than the maximum \(J_o\) when the hopper contains no obstacle. a2 Schematic defining a clogging flow regime where \(J_i>J_o\). b Averaged flow rates, \(J_i\) (orange) and \(J_a\) (red), normalized by \(J_o\) for the frictionless system with monodisperse disc particles and a round obstacle. Each data point is obtained using 18 different initial conditions. A zoomed-in plot at the bottom emphasizes the transition from \(J_i < J_o\) (fluidized flow regime, shaded) to \(J_i > J_o\) (clogging flow regime, unshaded), followed by the occurrence of a local peak of \(J_a\)

When the obstacle is located closer to the hopper orifice, \(J_i\) is lower than \(J_o\), defined as a fluidized flow regime as shown in Fig. 10a1, and we should observe a monotonic increase of the actual flow rate \(J_a\). On the other hand, when the obstacle is placed further away from the orifice, the two internal passages between the obstacle and the two hopper walls on its either side together can allow \(J_i\) to become higher than \(J_o\), defined as a clogging flow regime, as shown in Fig. 10a2. Presumably, \(J_a\) can be locally boosted in the clogging regime due to a greater-than-unity \(J_i/J_o\) that cannot be smoothly constrained by the hopper until the flow leaves its orifice, and therefore exhibits a local peak. \(J_a\) then increases again as the position H of the obstacle becomes higher until it eventually reaches its maximum \(J_o\). We believe that \(J_i/J_o>1\) is a necessary condition for observing a local flow rate peak.

To offer simulation evidence showing the proposed necessary condition is true, we plot \(J_o\), \(J_i\) and \(J_a\) of the frictionless case of monodisperse discs and a round obstacle as an example. The results are shown in Fig. 10b. To numerically measure \(J_i\), we put particles dropping below H back the top of the hopper but slightly lower than its top border by a distance of 0.1L. Additionally, we place a lid with a dimensionless damping parameter \(b^*_l=50b^*\) at the top border of the hopper to prevent fast-flying particles from escaping the simulation domain and conserve the total number of particles N in the system. As expected, we observe a monotonic increase of \(J_a\) while \(J_i\) is below \(J_o\). A peak of \(J_a\) occurs soon after \(J_i/J_o>1\), and therefore we validate the proposed necessary condition.

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Gao, GJ.J., Blawzdziewicz, J., Holcomb, M.C. et al. Understanding the local flow rate peak of a hopper discharging discs through an obstacle using a Tetris-like model. Granular Matter 21, 25 (2019).

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